Ratios, measurement, and probability

Ratios, averages, and measurement

Ratios

A ratio compares two or more quantities. A ratio of boys to girls of $3:2$ (read “3 to 2”) means that for every $3$ boys there are $2$ girls. The order matters: the terms follow the order the ratio names its quantities, so the ratio of girls to boys here is $2:3$. The numbers in a ratio are parts, and the parts add up to the whole.

Example:

A class of $30$ students has boys and girls in a $3:2$ ratio. How many are girls?

There are $3+2=5$ parts, so each part is $30÷5=6$ students.

Girls: $2\times 6=12$. (Boys: $3\times 6=18$.)

Answer: $12$ girls

The same method works for a ratio with three parts.

Example:

A jar holds red, white, and blue beads in a $2:3:5$ ratio. There are $12$ red beads. How many beads are in the jar?

A. $30$
B. $48$
C. $60$
D. $72$

The $2$ red parts hold $12$ beads, so one part is $12÷2=6$ beads. The jar holds $2+3+5=10$ parts:

$$10\times 6=60$$

Choice B counts only the white and blue beads ($8$ parts). Reread what the question asks for: the whole jar.

Answer: C

When the numbers change, work with the actual counts, not the ratio. If $6$ more boys join the class of $18$ boys and $12$ girls, there are $24$ boys and $12$ girls, a new ratio of $24:12$, or $2:1$.

Proportions and scale

A proportion says two ratios are equal. Set it up with the same kind of quantity in the same place on both sides, then cross-multiply.

Map and model scales are proportions. If $1$ inch on a map stands for $25$ miles, then $3.5$ inches stands for $3.5\times 25=87.5$ miles.

Knowledge check

A recipe uses a ratio of $3$ cups of water for every $2$ cups of juice. A large batch uses $18$ cups of water.

If the mixture is then poured equally into $5$ containers, how many cups are in each container?

A. $4$
B. $6$
C. $9$
D. $10$

(spoiler)

$18$ cups of water is $6$ times the recipe’s $3$ cups, so the batch uses $2\times 6=12$ cups of juice. The whole mixture is $18+12=30$ cups, and $30÷5=6$.

Answer: B. $6$

Averages

The average (or mean) is the sum of the values divided by how many values there are. Turned around, the sum is the average times the count, and that version solves most average word problems.

Example:

Jo’s first four test scores are $82$, $88$, $75$, and $91$. What score does Jo need on the fifth test to average $85$?

To average $85$ over $5$ tests, the scores must total $85\times 5=425$.

The first four total $82+88+75+91=336$.

$$425-336=89$$

Answer: $89$

Unit conversions

Memorize these common conversions:

• $1$ minute = $60$ seconds, and $1$ hour = $60$ minutes
• $1$ foot = $12$ inches, $1$ yard = $3$ feet, and $1$ mile = $5,280$ feet
• $1$ pound = $16$ ounces, and $1$ ton = $2,000$ pounds
• $1$ cup = $8$ fluid ounces, $1$ pint = $2$ cups, $1$ quart = $2$ pints, and $1$ gallon = $4$ quarts
• $1$ meter = $100$ centimeters, $1$ kilometer = $1,000$ meters, and $1$ kilogram = $1,000$ grams

To change to a smaller unit, multiply; to change to a larger unit, divide. Some problems take two steps: $2$ yards is $2\times 3=6$ feet, which is $6\times 12=72$ inches.

Square units convert by the square of the factor. A square yard is $3$ feet by $3$ feet, so it holds $3\times 3=9$ square feet. A room $12$ feet by $15$ feet is $180$ square feet, or $180÷9=20$ square yards of carpet. Dividing by $3$ instead of $9$ is the usual mistake.

Perimeter, area, and volume in word problems

Decide which measurement the situation calls for before you calculate:

• Perimeter is the distance around: fencing, trim, a border. A garden $8$ feet by $5$ feet needs $2\times (8+5)=26$ feet of fence.
• Area is the surface covered: carpet, paint, tile, sod. The same garden covers $8\times 5=40$ square feet.
• Volume is the space filled: water, sand, concrete. A box $4$ feet by $3$ feet by $2$ feet holds $4\times 3\times 2=24$ cubic feet.

Counting and probability

Counting choices

If one choice can be made in $m$ ways and a second in $n$ ways, the two together can be made in $m\times n$ ways. This is the counting principle, and it works for any number of choices.

Example:

Sam has $4$ pairs of pants, $6$ shirts, and $3$ belts. An outfit is one pair of pants, one shirt, and one belt. How many different outfits can Sam make?

$$4\times 6\times 3=72$$

Adding the counts ($4+6+3=13$) is the trap: each pair of pants can go with every shirt, and each of those combinations with every belt.

Answer: $72$ outfits

The number of ways to put $n$ different things in order is $n!$ (“$n$ factorial”): $n\times (n-1)\times \cdots \times 2\times 1$. Five people can line up for a photo in $5!=5\times 4\times 3\times 2\times 1=120$ different orders.

Probability

The goal of probability is to find the likelihood of an event happening out of all possible outcomes.

$$\frac{\text{event}}{\text{possibilities}}$$

Probabilities are always between $0$ and $1$.

• A probability of $0$ means an outcome is impossible
• A probability of $1$ means an outcome is guaranteed

We can also think of probability as a percentage:

• $0$ → $0\%$
• $1$ → $100\%$
• $\frac{1}{2}$ or $0.5$ → $50\%$

Example:

There are $6$ red marbles and $4$ blue marbles in a bag. What is the probability of pulling a blue marble?

Step 1: Find all possible outcomes. There are $6+4=10$ marbles, so $10$ is the denominator.

Step 2: Find how many times the event can occur. There are $4$ blue marbles:

$$\frac{4}{10}$$

Step 3: Simplify the fraction.

$$\frac{4}{10}=\frac{2}{5}$$

Answer: $\frac{2}{5}$, or $40\%$

When an event does not happen

An event either happens or it doesn’t, so the two probabilities add up to $1$:

$$P(\text{not }A)=1-P(A)$$

In a bag of $12$ marbles with $3$ green ones, the probability of drawing a green marble is $\frac{3}{12}=\frac{1}{4}$, so the probability of drawing one that is not green is $1-\frac{1}{4}=\frac{3}{4}$. You can also count directly: $9$ of the $12$ marbles aren’t green.

Two events

For two events that don’t affect each other, multiply their probabilities to find the chance that both happen. The probability that a coin lands heads and a die shows $6$ is:

$$\frac{1}{2}\times \frac{1}{6}=\frac{1}{12}$$

Adding the probabilities instead is a common mistake. Two things both happening is never more likely than either one alone, so the answer can’t be bigger than either probability.

When items are drawn without replacement, the first draw changes what’s left for the second.

Example:

A bag holds $4$ red and $6$ blue marbles. Two marbles are drawn without replacement. What is the probability that both are red?

A. $\frac{2}{15}$
B. $\frac{4}{25}$
C. $\frac{8}{45}$
D. $\frac{2}{5}$

First draw: $4$ of the $10$ marbles are red, so the probability is $\frac{4}{10}$.

Second draw: one red marble is gone, so $3$ of the remaining $9$ are red, a probability of $\frac{3}{9}$.

$$\frac{4}{10}\times \frac{3}{9}=\frac{12}{90}=\frac{2}{15}$$

Choice B, $\frac{4}{10}\times \frac{4}{10}$, is the answer if the first marble were put back.

Answer: A